Algebraic Curve -- from Wolfram MathWorld

An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .

The following table lists the names of algebraic curves of a given degree.

order	curve	examples
2	quadratic curve	circle, ellipse, hyperbola, parabola
3	cubic curve	cissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi
4	quartic curve	ampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid of Nicomedes, cruciform, deltoid, devil's curve, Dürer's conchoid, eight curve, fish curve, hippopede, Kampyle of Eudoxus, Kepler's folium, Klein quartic, knot curve, lemniscate, limaçon, links curve, pear-shaped curve, piriform curve, swastika curve, trefoil curve, trifolium
5	quintic curve	Burnside curve, butterfly catastrophe curve, stirrup curve
6	sextic curve	astroid, atriphtaloid, Cayley's sextic, cornoid, cycloid of Ceva, dumbbell curve, ellipse evolute, epicycloid, Freeth's nephroid, heart curve (first), limaçon evolute, nephroid, quadrifolium, scarabaeus curve, Talbot's curve
8	octic curve	pear curve
12	dodecic curve	ranunculoid

References

Arbarello, E.; Cornalba, M.; Griffiths, P. A.; and Harris, J. Geometry of Algebraic Curves, I. New York: Springer-Verlag, 1985.

Griffiths, P. A. Introduction to Algebraic Curves. Providence, RI: Amer. Math. Soc., 1989.

Algebraic Curve

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